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Lecture 9: Benchmark tracking problems

Prof. Dr. Svetlozar Rachev

Institute for Statistics and Mathematical EconomicsUniversity of Karlsruhe

Portfolio and Asset Liability Management

Summer Semester 2008

Prof. Dr. Svetlozar Rachev (University of Karlsruhe)Lecture 9: Benchmark tracking problems 2008 1 / 62

Copyright

These lecture-notes cannot be copied and/or distributed withoutpermission.The material is based on the text-book:Svetlozar T. Rachev, Stoyan Stoyanov, and Frank J. FabozziAdvanced Stochastic Models, Risk Assessment, and PortfolioOptimization: The Ideal Risk, Uncertainty, and PerformanceMeasuresJohn Wiley, Finance, 2007

Prof. Svetlozar (Zari) T. RachevChair of Econometrics, Statisticsand Mathematical FinanceSchool of Economics and Business EngineeringUniversity of KarlsruheKollegium am Schloss, Bau II, 20.12, R210Postfach 6980, D-76128, Karlsruhe, GermanyTel. +49-721-608-7535, +49-721-608-2042(s)Fax: +49-721-608-3811http://www.statistik.uni-karslruhe.de


Introduction

An important problem for fund managers is comparing the performance oftheir portfolios to a benchmark. The benchmark could be:

a market index or

any other portfolio or

liability measure in the case of defined benefit pension plans.

There are two types of strategies that managers follow: active or passive.

An active portfolio strategy uses available information and forecastingtechniques to seek a better performance. It is an expectation about thefactors that could influence the performance of an asset class. The goalof an active strategy is to outperform the benchmark after managementfees by a given number of basis points.

A passive portfolio strategy involves minimal expectational input andinstead relies on diversification to match the performance of somebenchmark. A passive strategy, commonly referred to as indexing,assumes that the marketplace will reflect all available information in theprice paid for securities.


Introduction

There are various strategies for constructing a portfolio toreplicate the index but the key in these strategies is designing aportfolio whose tracking error relative to the benchmark is as smallas possible.

Tracking error is the standard deviation of the difference betweenthe return on the replicating portfolio and the return on thebenchmark.

⇒ The benchmark tracking problem can be formulated as anoptimization problem.


Introduction

We’ll consider the benchmark tracking problem from a verygeneral viewpoint, replacing the traditional tracking errormeasures by a general functional satisfying a number of axioms.

We call this functional a metric of relative deviation because itcalculates the relative performance of the portfolio to thebenchmark.

Our approach is based on the universal methods of the theory ofprobability metrics. As a result, the optimization problems whicharise are a significant generalization of the currently existingapproaches to benchmark tracking.


The tracking error problem

In lecture 8, the optimal portfolio problems have one feature incommon in that the risk measure, or the dispersion measure,concerns the distribution of portfolio returns without any referenceto another portfolio.

In contrast, benchmark-tracking problems include a benchmarkportfolio against which the performance of the managed portfoliowill be compared.

The arising optimization problems include the distribution of theactive portfolio return defined as the difference rp − rb in which rp

denotes the return of the portfolio and rb denotes the return of thebenchmark.

If the active return is positive, this means that the portfoliooutperformed the benchmark and, if the active return is negative,then the portfolio underperformed the benchmark.



In the ex-post analysis, we observe a specified historical period intime and try to evaluate how successful the portfolio manager wasrelative to the benchmark.

In this case, there are two time series corresponding to theobserved portfolio returns and the observed benchmark returns.

A measure of the performance of the portfolio relative to thebenchmark is the average active return, also known as theportfolio alpha and denoted by αp.



Alpha is calculated as the difference between the average of theobserved portfolio returns and the average of the observedbenchmark returns,

α̂p = r̄p − r̄b,

where

α̂p denotes the estimated alphar̄p = 1

k

∑ki=1 rpi denotes the average of the observed

portfolio returns rp1, rp2, . . . , rpk

r̄b = 1k

∑ki=1 rbi denotes the average of the observed

benchmark returns rb1, rb2, . . . , rbk



A widely used measure of how close the portfolio returns are tothe benchmark returns is the standard deviation of the activereturn, also known as tracking error.

When it is calculated using historical observations, it is referred toas the ex-post or backward-looking tracking error.

If the portfolio returns are equal to the benchmark returns in thespecified historical period, rpi = rbi for all i , then the observedactive return is equal to zero and, therefore, the tracking error willbe equal to zero.

⇒ The closer the tracking error to zero, the closer the risk profile of theportfolio matches the risk profile of the benchmark.



In the ex-ante analysis, portfolio alpha equals the mathematicalexpectation of the active return,

αp = E(rp − rb)

= w ′µ − Erb,(1)

where rp = w ′X in which w denotes the vector of portfolio weights,X is a random vector describing the future assets returns, andµ = EX is a vector of the expected assets returns.

The tracking error equals the standard deviation of the activereturn,

TE(w) = σ(rp − rb),

where σ(Y ) denotes the standard deviation of the r.v. Y .

⇒ Tracking error in this case is referred to as ex-ante orforward-looking tracking error.



If the strategy followed is active, then the goal of the portfoliomanager is to gain a higher alpha at the cost of deviating from therisk profile of the benchmark portfolio; that is, the manager willaccept higher forward-looking tracking error.

⇒ Active strategies are characterized by high alphas and highforward-looking tracking errors.

If the strategy is passive, then the general goal is to construct aportfolio so as to have a forward-looking tracking error as small aspossible in order to match the risk profile of the benchmark. As aconsequence, the alpha gained is slightly different from zero.

⇒ Passive strategies are characterized by very small alphas and verysmall forward-looking tracking errors.



Strategies which are in the middle between the active and thepassive ones are called enhanced indexing.1

In following such a strategy, the portfolio manager constructs aportfolio with a risk profile close to the risk profile of thebenchmark but not identical to it.

Enhanced indexing strategies are characterized by small tomedium-sized forward-looking tracking errors and small tomedium-sized alphas.

The optimal portfolio problem originating from this framework isthe minimal tracking error problem.

1Loftus (2000) classifies the three strategies for equity portfolio strategies asfollows: indexing, 0 to 20 basis points; 50 to 200 basis points; and, activemanagement, 400 basis points and greater.



The minimal tracking error problem has the following form

minw

σ(rp − rb)

subject to w ′e = 1w ′µ − Erb ≥ R∗

w ≥ 0,

(2)

where R∗ denotes the lower bound of the expected alpha.

Its structure is very similar to the mean-variance optimizationproblems. The difference is that the active portfolio return rp − rb isused instead of the absolute portfolio return rp.

The goal is to find a portfolio which is closest to the benchmark ina certain sense, while setting a threshold on the expected alpha.In this case, the “closeness” is determined by the standarddeviation.



By varying the limit R∗, we obtain the entire spectrum frompassive strategies (R∗ close to zero), through enhanced indexing(R∗ taking medium-sized values), to active strategies (R∗ takingfrom medium-sized to large values).

The set of the optimal portfolios generated by problem (2) is theset of efficient portfolios which, if plotted in the plane of expectedalpha versus tracking-error, form the corresponding efficientfrontier. (See Figure 1 for illustration.)



Tracking error

Exp

ecte

d al

pha

Tracking error0

Enhanced indexing Active strategies

Figure 1. The efficient frontier generated from the minimal trackingerror problem. The passive strategies are positioned to the left ofenhanced indexing strategies.



If the investment universe is the same as or larger than theuniverse of the benchmark portfolio, then the global minimumtracking error is equal to zero. Then the optimal portfolio coincideswith the benchmark portfolio.

The global minimum tracking error portfolio represents a typicalpassive strategy.

Increasing the lower bound on the expected alpha we enter thedomain of enhanced indexing.

Increasing further R∗ leads to portfolios which can becharacterized as active strategies.



Recall that a serious disadvantage of the standard deviation isthat it penalizes in the same way the positive and the negativedeviations from the mean of the r.v.

Then the tracking error treats in the same fashion theunderperformance and the outperformance, while our attitudetowards them is asymmetric.

We are inclined to pay more attention to the underperformance.

⇒ From an asset management perspective, a more realistic measureof “closeness” should be asymmetric.



Our aim is to restate the minimal tracking error problem in themore general form

minw

µ(rp, rb)

subject to w ′e = 1w ′µ − Erb ≥ R∗

w ≥ 0,

(3)

where µ(X , Y ) is a measure of the deviation of X relative to Y .

Due to this interpretation, we regard µ as a functional whichmetrizes relative deviation and we call it a relative deviation metricor simply, r.d. metric.



If the portfolio rp is an exact copy of the benchmark, i.e. it containsexactly the same stocks in the same amounts, then the relativedeviation of rp to rb should be close to zero.2

The converse should also hold but, generally, could be in a somewhatweaker sense.

If the deviation of rp relative to rb is zero, then the portfolio and thebenchmark are indistinguishable but only in the sense implied byµ. They may, or may not, coincide with probability 1.

2It would not be equal to zero due to transaction costs.Prof. Dr. Svetlozar Rachev (University of Karlsruhe)Lecture 9: Benchmark tracking problems 2008 19 / 62

Relation to probability metrics

The benchmark-tracking problem given by (3) belongs to a classof problems in which distance between random quantities ismeasured.

In order to gain more insight into the properties that µ shouldsatisfy, we relate the benchmark-tracking problem to the theory ofprobability metrics, explained in Lecture 3.



A functional which measures the distance between randomquantities is called a probability metric.

These random quantities can be random variables (such as thedaily returns of equities, the daily change of an exchange rate,etc.) or stochastic processes (such as a price evolution in a givenperiod), or much more complex objects (such as the dailymovement of the shape of the yield curve).

The probability metric is defined through a set of axioms, given inLecture 3; that is, any functional which satisfies these axioms iscalled a probability metric.



From the standpoint of the theory of probability metrics, thebenchmark-tracking problem given by (3) can be viewed as anapproximation problem.

So that we are trying to find a r.v. rb in the set of feasible portfolioswhich is closest to the r.v. rb and the distance is measured by thefunctional µ.

This functional should satisfy the properties stated in Lecture 3, orsome versions of them, in order for the problem to give meaningfulresults.

Let us reexamine the set of properties Property 1 - Property 3 toverify if some of them can be relaxed while application to aspecific problem.



Property 1 and Property 3, we leave intact. The reason is thatanything other than Property 1 is just nonsensical and Property 1together with Property 3 guarantee nice mathematical properties,such as continuity of µ. Property 3 alone makes sense because ofthe interpretation that we are measuring distance.

Property 2 can be dropped. The rationale is that, in problem (3)the assumption of asymmetry is a reasonable property because ofour natural tendency to be more sensitive to underperformancethan to outperformance relative to the benchmark portfolio.



The nature of the problem may require the additional properties.

Let us consider two equity portfolios with returns X and Y .

Suppose that we convert proportionally into cash 100 a % in totalof both portfolios where 0 ≤ a ≤ 1 stands for the weight of thecash amount.

As a result, the two portfolios returns scale down to (1 − a)X and(1 − a)Y respectively.

Since both random quantities get scaled down by the same factor,we may assume that the distance between them scales downproportionally. Actually, we assume that the distance scales downby the same factor raised by some fixed power s,

ν(aX , aY ) = asν(X , Y ) for any X , Y and a,≥ 0.

If s = 1, then the scaling is proportional.



The reason we are presuming a more general property is thatdifferent classes of r.d. metrics will originate and depending on sthey may have different robustness in the approximation problem.

This property we call positive homogeneity of degree s. It issimilar to the homogeneity property of ideal probability metrics.



As a next step, consider an equity with return Z which isindependent of the two portfolios returns X and Y .

Suppose that we invest the cash amounts into equity Z . Thereturns of the two portfolios change to (1 − a)X + aZ and(1 − a)Y + aZ , respectively, where 0 ≤ a ≤ 1 denotes the weightof equity Z in the portfolio.

How does the distance change?

Certainly, there is no reason to expect that the distance shouldincrease. It either remains unchanged or decreases.



In terms of the functional, we assume the property

ν(X + Z , Y + Z ) ≤ ν(X , Y )

for all Z independent of X , Y .

Any functional ν satisfying this property, we call weakly regular, alabel we borrow from the probability metrics theory. In fact, this isthe weak regularity property of ideal probability metrics.

If the distance between the two new portfolios remains unchangedfor any Z irrespective of the independence hypothesis, then wesay that ν is translation invariant.



Note that if the positive homogeneity property and the weakregularity property hold, then the inequality

ν((1 − a)X + aZ , (1 − a)Y + aZ ) ≤ ν(X , Y ), a ∈ (0, 1) (4)

holds as well and this is exactly the mathematical expressionbehind the conclusion in the example.

While the weak regularity property may seem more confined thanpostulating directly (4), we assume it as an axiom because (4) istied to the interpretation of the random variables as return oninvestment.

Suppose that this is not the case and X , Y denote the randomwealth of the two portfolios under consideration and Z denotesrandom stock price.



Furthermore, assume that the present value of both portfolios areequal and that we change both portfolios by buying one share ofstock Z .

Then the random wealth of both portfolios becomes X + Z andY + Z , respectively, and, because of the common stochastic factorZ , we expect the relative deviation to decrease; that isν(X + Z , Y + Z ) ≤ ν(X , Y ).

⇒ The weak regularity property is the fundamental property we wouldlike to impose.



In order to state the last axiom, suppose that we add to the twoinitial portfolios other equities, such that returns of the portfoliosbecome X + c1 and Y + c2, where c1 and c2 are some constants.

We assume that the distance between the portfolios remainsunchanged because it is only the location of X and Y thatchanges. That is,

ν(X + c1, Y + c2) = ν(X , Y )

for all X , Y and constants c1, c2. We call this property locationinvariance.

As a corollary, this property allows measuring the distance onlybetween the centered portfolios returns.



We demonstrate how such a functional ν can be constructed, forexample, from a given probability metric.

Suppose that µ(X , Y ) is a given probability metric and denote byg the mapping

g : X → X − EX .

The mapping takes as an argument a random variable with a finitemean and returns as output the random variable with its meansubtracted, g(X ) = X − EX .

The mapping g has the property that shifting the random variableX does not change the output of the mapping,

g(X + a) = X − EX = g(X ).



Let us define the functional ν as

ν(X , Y ) = µ(g(X ), g(Y )). (5)

Thus, ν calculates the distance between the centered random variablesX − EX and Y − EY by means of the probability metric µ.

As a consequence, the functional ν defined in (5) is location invariant,

ν(X + c1, Y + c2) = µ(g(X + c1), g(Y + c2))

= µ(g(X ), g(Y ))

= ν(X , Y ).

The definition in equation (5) can be written in a more compact formwithout introducing an additional notation for the mapping,

ν(X , Y ) = µ(X − EX , Y − EY ).



The expected return of the portfolio and some othercharacteristics can be incorporated into the constraint set of thebenchmark-tracking problem (3).

For example, the expected alpha constraint imposes a lowerbound on the expected alpha, or the expected outperformancerelative to the benchmark.



Let’s finally define the r.d. metrics.

Any functional µ which is weakly regular, location invariant, positivelyhomogeneous of degree s, and satisfies Property 1 and Property 3.

The structural classification of probability metrics holds for r.d. metricsas well. We distinguish between compound, simple, and primary r.d.metrics depending on the degree of sameness implied by the r.d.metric.



Now let us revisit the classical tracking error and try to classify it.

First, it is a special case of the average compound metric withp = 2 and therefore it satisfies Property 1 - Property 3. Of course,this also means it is a compound metric, hence it implies thestrongest form of sameness — in almost sure sense.

Second, concerning the group of the additional axioms, it ispositively homogeneous of degree 1, translation invariant, andsatisfies the location invariance property.

Further, our goal is to give other examples of r.d. metrics, which aresubstantially different from classical tracking error, and to see theirproperties in a practical setting.


Examples of r.d. metrics

A deviation measure D(X ) can generate a functional which is areasonable candidate for a measure of distance in theoptimization problem (3).

For example,µD(X , Y ) = D(X − Y ) (6)

is a translation invariant probability semimetric, homogeneous ofdegree 1 on condition that D is a symmetric deviation measure.

A converse relation holds as well. That is,

Dµ(X ) = µ(X − EX , 0) (7)

is a symmetric deviation measure, where µ is a translationinvariant probability metric, homogeneous of degree 1.



If D is general deviation measure, then µD is a r.d. metric. In asimilar way, if µ is a r.d. metric, then Dµ is a general deviationmeasure3.

⇒ All deviation measures turn out to be spawned from the class oftranslation invariant r.d. metrics with degree of homogeneity s = 1.

This relationship already almost completely classifies thefunctional µD arising from the deviation measure D.

Note that µD is a compound metric and therefore it implies thestrongest, almost sure sense of similarity. This can be seen byconsidering an example in which X and Y are independent andidentically distributed. Then the difference X − Y is a r.v. withnon-zero uncertainty, hence D(X − Y ) > 0.

3See the appendix of this lecture for the detailsProf. Dr. Svetlozar Rachev (University of Karlsruhe)Lecture 9: Benchmark tracking problems 2008 37 / 62


We proceed by providing two r.d. metrics belonging to a completelydifferent category - they both are simple and therefore the sense ofsimilarity they imply is only up to equality of distribution functions.

These functionals are defined through the equations

θ∗

p(X , Y ) =

[∫

∞

−∞

(max(FX (t) − FY (t), 0))pdt]1/p

, p ≥ 1 (8)

and

ℓ∗p(X , Y ) =

[

∫ 1

0(max(F−1

Y (t) − F−1X (t), 0))pdt

]1/p

, p ≥ 1 (9)

where

X and Y are zero-mean random variables,FX (t) = P(X < t) is the distribution function of X andF−1

X (t) = inf{x : FX (x) ≥ t} is the generalized inverse of the distributionfunction.



The intuition behind (9) and (8) is the following.

Suppose that X and Y represent the centered random return oftwo portfolios and that their distribution functions are as shown inFigure 2.

Both functionals measure the relative deviation of X and Y usingonly the part of the distribution functions, or the inversedistribution functions, which describes losses.



−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

t

F(t

)

FX(t)

FY(t)

0 0.2 0.4 0.6 0.8 1−3

−2

−1

0

1

2

3

t

F−

1 (t)

FX−1(t)

FY−1(t)

Figure 2. The distribution functions (left) and the inverse distributionfunctions (right) of X and Y .



A closer look at the left plot in the figure reveals that the differenceFX (t) − FY (t) is non-negative only for negative values of t andtherefore θ

∗

p(X , Y ) essentially uses the information about lossescontained in the distribution function. The same holds for the otherfunctional.

In the case where p = 1, then θ∗

p(X , Y ) calculates the areabetween the two distribution functions to the left of the origin,which is exactly the same area between the inverse distributionfunctions to the left of t = 1/2.

It is easy to notice that θ∗

1(X , Y ) = ℓ∗1(X , Y ) but this is, generally,not true if p 6= 1.



A remark for (8) and (9) on Property 1 follows as there is a subtlenuance which has to be explained.

If the two distribution functions coincide, then both (9) and (8)become equal to zero.

The converse statement holds as well. Suppose that the two r.d.metrics are equal to zero. Then, the distribution functions of therandom variables may diverge but only in a very special way,

θ∗

p(X , Y ) = 0 =⇒ FX (t) ≤ FY (t), ∀t ∈ R.

See the illustration on the next slide.



−3 −2 −1 0 1 2 30

0.2

0.4

0.6

0.8

1

t

F(t

)

FX(t)

FY(t)

0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5

2

2.5

t

F−

1 (t)

FX−1(t)

FY−1(t)

Figure 3. The distribution functions (left) and the inverse distributionfunctions (right) of X and Y .



However, the inequality is impossible to hold for r.d. metricsbecause of the location invariance property; that is, we consideronly zero-mean random variables and the inequality between thedistribution functions above implies that EX ≤ EY , hence one ofthe random variables may have a non-zero mean.

As a result, if θ∗p(X , Y ) = 0, then it follows that the c.d.f.s coincidefor all values of the argument, FY (t) = FX (t), ∀t ∈ R and,therefore, the two random variables have identical probabilisticproperties.

Exactly the same argument applies to ℓ∗p(X , Y ).



In summary, we showed that Property 1 holds in a stronger sensefor (9) and (8).

Not only does the distance between X and X equal zero,µ(X , X ) = 0, but if µ(X , Y ) = 0, then for these two cases, becauseof the location invariance property, it follows that X is equivalent toY to the extent that their distribution functions coincide.

However, there are examples of r.d. metrics for which the locationinvariance property is insufficient to guarantee this strongeridentity property.



Is is possible to calculate explicitly the r.d. metrics (9) and (8)?

The answer is negative but in some special cases, this can be done.

Suppose that p = 1 and that both random variables have the normaldistribution, X ∈ N(0, σ2

X ) and Y ∈ N(0, σ2Y ).

Due to the equality θ∗

1(X , Y ) = ℓ∗1(X , Y ), it makes no difference whichr.d. metric we choose for this calculation. Then, under theseassumptions,

ℓ1(X , Y ) =

∫ 1

0(max(F−1

Y (t) − F−1X (t), 0))dt

=1√2π

|σX − σY | .(10)

In this special case, (10) is actually a primary metric because itmeasures the distance between the standard deviations of the portfolioreturn and the benchmark return. It is because we have restricted ourreasoning to the normal distribution only that the simple metric ℓ1 takesthis special form. Otherwise, it is a simple r.d. metric.



Looking more carefully at (10), we notice that the symmetryproperty holds due to the absolute value; that is, in this specialcase, ℓ1(X , Y ) = ℓ1(Y , X ).

This may appear striking because equation (9) is asymmetric byconstruction, or so it may seem.

The symmetry property appears, again, because of the normalityassumption — the left and the right tails of the distributionsdisagree symmetrically in this case.

In other words, the particular form of (9) allows for asymmetry ifthe corresponding distributions are skewed. If X and Y aresymmetric, then this is a fundamental limitation and the potentialfor asymmetry, granted by the r.d. metric, cannot be exploited.



We can use equation (10) to illustrate a point about the relationshipbetween the compound r.d. metrics and the minimal r.d. metricscorresponding to them.

Using very general arguments, only the triangle inequality, we can showthat equation (10) is related to the tracking error through the inequality

|σX − σY | ≤ σ(X − Y ). (11)

It is true not only when X and Y are normal but in general.

Equation (11) shows that if the tracking error is zero, then |σX − σY | = 0which, in the normal distribution case, means that ℓ1(X , Y ) = 0.

Conversely, in the normal distribution case one can find two randomvariables X and Y with σX = σY and, yet, the tracking error can benon-zero, σ(X − Y ) 6= 0, because of the dependence between the tworandom variables.



ExampleIf X and Y are independent, then X − Y is a r.v. with the normaldistribution and its standard deviation is strictly positive.

ℓ1(X , Y ) = 0 = |σX − σY | means that X and Y have the sameprobabilistic properties and, nevertheless, the tracking error maybe strictly positive.

Similar conclusion holds in general, when we consider compoundversus simple metrics but an inequality such as (11) is guaranteedto hold between compound metrics and the minimal metricscorresponding to them.

It is in the normal distribution case that the left side of (11)coincides with the minimal metric of the tracking error.


Numerical example

We showed that both functionals (9) and (8) are meaningfulobjectives in the benchmark-tracking problem. They are verydifferent from the classical tracking error as they are simplemetrics and imply a weaker form of sameness.

Even if they are both simple, the optimal solutions correspondingto (9) and (8) will, generally, not be the same if p 6= 1. This isunderstandable as the functionals are not identical.

There is one important difference between them concerningProperty 4. The functional θ

∗

p(X , Y ) is positively homogeneous ofdegree 1/p while ℓ∗p(X , Y ) is positively homogeneous of degree 1irrespective of the value of p.


Numerical example

We’ll provide a numerical example. Our goals are:1 Observe the difference between the optimal solutions of the

classical tracking error on the one hand and (9) and (8) on theother. In this example, the optimal solution is represented by aportfolio, the empirical c.d.f. of which is closest to the empiricalc.d.f. of the benchmark as measured by the corresponding r.d.metric.

2 Examine the effect of the degree of homogeneity in the case ofθ∗

p(X , Y ), our expectation being that the higher the degree ofhomogeneity, the more sensitive θ

∗

p(X , Y ) is.

Our dataset includes 10 randomly selected equities from the S&P 500universe and the benchmark is the S&P 500 index. The data cover theone-year period from December 31, 2002 to December 31, 2003.


Numerical example

The optimization problem we solve is the benchmark-trackingproblem given by (3) in which R∗ = 0 and the objective functionµ(rp, rb) is represented by the corresponding empiricalcounterpart,

σ̂(rp − rb) (12)

θ̂∗

p(rp0, rb0) (13)

ℓ̂∗p(rp0, rb0) (14)

where the index 0 signifies that the corresponding returns arecentered.

In the case of tracking error, the sample counterpart σ̂ is thesample standard deviation.


Numerical example

In the three problems, we start from an equally weighted portfolioand then solve the optimization problems.

Therefore, in all cases, our initial portfolio is an equally weightedportfolio of the 10 randomly selected stocks.

The constraint set guarantees that the expected return of theoptimal portfolio will not be worse than that of the benchmark.

We compare the inverse distribution functions of the centeredreturns of the optimal portfolios in order to assess which problembetter tracks the benchmark in terms of the distribution function.

Note that it makes no difference whether we compare thedistribution functions or the inverse distribution functions, theconclusion will not change.


Numerical exampleFigure below compares the inverse distribution functions of the centered returns of theinitial portfolio, the optimal portfolio of the classical tracking error problem and theoptimal portfolio obtained with objective (14) in which p = 1.

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t

F−

1 (t)

Initial PortfolioBenchmark

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t

F−

1 (t)

Classical tracking error

Optimal PortfolioBenchmark

Figure 4. The inverse distribution functions of the S&P 500 index (the benchmark),equally weighted portfolio (initial portfolio) and the two optimal portfolios.


Numerical example

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t

F−1(t)

l1* (r

p0, r

b0)


Figure 4 (cont.). The inverse distribution functions of the S&P 500index (the benchmark), equally weighted portfolio (initial portfolio) andthe two optimal portfolios.


Numerical example

It is obvious that both optimization problems provide solutions that bettertrack the benchmark than the trivial strategy of holding an equallyweighted portfolio.

The functional (14) does a better job at approximating the distributionfunction of the benchmark returns, allowing for asymmetries in the lossversus the profit part.

The part of the inverse distribution function of the optimal solutiondescribing losses, the one closer to t = 0 is closer to the correspondingpart of the inverse distribution function of the benchmark returns, whilethis is not true for the profit part closer to t = 1.

Actually, the fact that the inverse distribution function of the optimalsolution is above the inverse distribution function of the benchmarkreturns close to t = 1 means that the probability of a large positive returnof the optimal portfolio is larger than that of the benchmark.


Numerical example

In order to explore the question of how the degree of homogeneitymight influence the solution, we solve the tracking problem withobjective (13) in which we choose p = 1 and p = 10.

The degree of homogeneity is equal to 1 and 1/10 respectively.

We noted already that θ∗

1(rp0, rb0) = ℓ∗1(rp0, rb0) and therefore theoptimal solutions will coincide.

The inverse distribution functions of the returns of the optimalportfolios are shown on Figure 5.


Numerical example

0 0.2 0.4 0.6 0.8 1−0.06

−0.04

−0.02

0

0.02

0.04

0.06

t

F−1(t)

θ10* (r

p0, r

b0)


Figure 5. The inverse distribution functions of the SP500 index (thebenchmark), and the optimal portfolios obtained with (13) as objectivewith p = 10.


Numerical example

Apparently, the degree of sensitivity of the objective is directlyinfluenced by the parameter p in line with our expectations.

The integrand in the functional (13) measures the differences ofthe two distribution functions and therefore its functional valuesare small numbers, converging to zero in the tails.

Holding other things equal, raising the integrand to a higher powerdeteriorates the sensitivity of the functional with respect todeviations in the tails of the two distribution functions.

This observation becomes obvious when we compare the bottomplot of Figure 4 to Figure 5.


Numerical example

Generally, the optimization problems involving simple r.d. metricsmay not belong to the family of convex problems because thesimple r.d. metric may not appear to be a convex function ofportfolio weights.

Stoyanov, Rachev and Fabozzi (2007) show that, in particular, thisholds for the minimal r.d. metrics.


Summary

We considered the problem of benchmark-tracking.

The classical problem relies on the tracking error to measure thedegree of similarity between the portfolio and the benchmark.

Making use of the approach of the theory of probability metrics,we extended significantly the framework by introducingaxiomatically relative deviation metrics replacing the tracking errorin the objective function of the optimization problem.

We provided two examples of relative deviation metrics and anumerical illustration of the corresponding optimization problems.


Svetlozar T. Rachev, Stoyan Stoyanov, and Frank J. FabozziAdvanced Stochastic Models, Risk Assessment, and PortfolioOptimization: The Ideal Risk, Uncertainty, and PerformanceMeasuresJohn Wiley, Finance, 2007.

Chapter 9.


Prof. Dr. Svetlozar Rachev€¦ · Prof. Dr. Svetlozar Rachev Institute for Statistics and...

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